148 THE FACTORS OF THE MIND

hypothesis we could adopt either of the criteria proposed by Yule,
e.g, calculate ratios for the proportionality criterion ([25], p. 28,

or, what amounts to the same thing, calculate the differences
between the * cross-products ' in all the available * tetrads/ as he
terms them. " In the case of complete * independence ' the
association is evidently zero for every tetrad " (i.e. map . fn^ —
Wbp • w>aq = o : [25], p. 69) .* Systematically carried out, either
procedure is really a test for a matrix of rank one.2

The precisian, however, will now inquire whether a
* better fit ' could not be obtained by weighting the tests
before they are averaged. But what weights are we to
choose ? In correlational work " the term ' best fit ? is
used as in the method of least squares : a c best-fit ' deter-
mination will therefore be one in which the sum of the
squares of the deviations is a minimum, i.e. the standard
error of estimate is a minimum" ([47]? p. 159)- If we
accept this convention for factorial work as well, we have
at once a well-established principle on which to base a more
exact procedure ; and, as we shall see later on, what may
be called the ' least-squares method ' of factor-analysis
yields at once, by an easy calculation, a suitable set of weights
and proves to be widely applicable, not only for the case of
a single factor, but also for the case of many ([93], p. 247).

As in other problems where the principle of least squares
is applied, these considerations lead us directly to the matrix
of product-sums or covariances. Now, if the measurements
matrix has, actually or in theory, a rank of only one, then
the matrix of covariances must also have a rank of only one.
This is obvious: for, since Jkf =£.p and p.p' — l?
R =AfM'=l.f.

We thus reach what may be called the single-factor
theorem for a covariance or correlation matrix. " Any
symmetric table, in which the rows (and therefore also the

1 Yule's methods were first described in his paper * On the Association of
Attributes/ Phil Trans. Roy, Soc., A, CXCIV, 1900, p. 257 f.

* It appears as such, in the chapter on the elementary theory of determin-
ants in most books on algebra.